# ifndef CPPAD_EXAMPLE_ABS_NORMAL_QP_BOX_HPP
# define CPPAD_EXAMPLE_ABS_NORMAL_QP_BOX_HPP
/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-17 Bradley M. Bell

CppAD is distributed under multiple licenses. This distribution is under
the terms of the
                    Eclipse Public License Version 1.0.

A copy of this license is included in the COPYING file of this distribution.
Please visit http://www.coin-or.org/CppAD/ for information on other licenses.
-------------------------------------------------------------------------- */
/*
$begin qp_box$$
$spell
	hpp
	const
	col
	xout
	yout
	sout
	cols
	prog
	maxitr
	xin
	qp
$$
$section abs_normal: Solve a Quadratic Program With Box Constraints$$

$head Syntax$$
$icode%ok% = qp_box(
	%level%, %a%, %b%, %c%, %C%, %g%, %G%, %epsilon%, %maxitr%, %xin%, %xout%
)%$$

$head Prototype$$
$srcfile%example/abs_normal/qp_box.hpp%
	0%// BEGIN PROTOTYPE%// END PROTOTYPE%
1%$$

$head Source$$
This following is a link to the source code for this example:
$cref/qp_box.hpp/qp_box.hpp/$$.

$head Purpose$$
This routine could be used to create a version of $cref abs_min_linear$$
that solved quadratic programs (instead of linear programs).

$head Problem$$
We are given
$latex a \in \B{R}^n$$,
$latex b \in \B{R}^n$$,
$latex c \in \B{R}^m$$,
$latex C \in \B{R}^{m \times n}$$,
$latex g \in \B{R}^n$$,
$latex G \in \B{R}^{n \times n}$$,
where $latex G$$ is positive semi-definite.
This routine solves the problem
$latex \[
\begin{array}{rl}
\R{minimize} &
\frac{1}{2} x^T G x + g^T x \; \R{w.r.t} \; x \in \B{R}^n
\\
\R{subject \; to} & C x + c \leq 0 \; \R{and} \; a \leq x \leq b
\end{array}
\] $$
The matrix $latex G + C^T C$$ must be positive definite on components
of the vector $latex x$$ where the lower limit minus infinity
and the upper limit is plus infinity; see $icode a$$ and $icode b$$ below.

$head Vector$$
The type $icode Vector$$ is a
simple vector with elements of type $code double$$.

$head level$$
This value is less that or equal two.
If $icode%level% == 0%$$,
no tracing is printed.
If $icode%level% >= 1%$$,
a trace of the $code qp_box$$ operations is printed.
If $icode%level% == 2%$$,
a trace of the $cref qp_interior$$ sub-problem is printed.

$head a$$
This is the vector of lower limits for $latex x$$ in the problem.
If $icode%a%[%j%]%$$ is minus infinity, there is no lower limit
for $latex x_j$$.

$head b$$
This is the vector of upper limits for $latex x$$ in the problem.
If $icode%a%[%j%]%$$ is plus infinity, there is no upper limit
for $latex x_j$$.

$head c$$
This is the value of the inequality constraint function at $latex x = 0$$.

$head C$$
This is a $cref/row-major/glossary/Row-major Representation/$$ representation
of thee the inequality constraint matrix $latex C$$.

$head g$$
This is the gradient of the objective function.

$head G$$
This is a row-major representation of the Hessian of the objective function.
For $latex j = 0 , \ldots , n-1$$,
$latex - \infty < a_j$$ or
$latex b_j < + \infty$$ or
$latex G_{j,j} > 0.0$$.

$head epsilon$$
This argument is the convergence criteria;
see $cref/KKT conditions/qp_box/KKT Conditions/$$ below.
It must be greater than zero.

$head maxitr$$
This is the maximum number of
$cref qp_interior$$ iterations to try before giving up
on convergence.

$head xin$$
This argument has size $icode n$$ and is the initial point for the algorithm.
It must strictly satisfy the constraints; i.e.,
$codei%
	%a% < %xin%,  %xin% < %b%,  %C% * %xin% - %c% < 0
%$$

$head xout$$
This argument has size is $icode n$$ and
the input value of its elements does no matter.
Upon return it is the primal variables
$latex x$$ corresponding to the problem solution.

$head ok$$
If the return value $icode ok$$ is true, convergence is obtained; i.e.,
$latex \[
	| F ( x , y_a, s_a, y_b, s_b, y_c, s_c ) |_\infty < \varepsilon
\] $$
where $latex |v|_\infty$$ is the infinity norm of the vector $latex v$$,
$latex \varepsilon$$ is $icode epsilon$$,
$latex x$$ is equal to $icode xout$$,
$latex y_a, s_a \in \B{R}_+^n$$,
$latex y_b, s_b \in \B{R}_+^n$$ and
$latex y_c, s_c \in \B{R}_+^m$$.

$head KKT Conditions$$
Give a vector $latex v \in \B{R}^m$$ we define
$latex D(v) \in \B{R}^{m \times m}$$ as the corresponding diagonal matrix.
We also define $latex 1_m \in \B{R}^m$$ as the vector of ones.
We define
$latex \[
F ( x , y_a, s_a, y_b, s_b, y_c, s_c )
=
\left(
\begin{array}{c}
g + G x - y_a + y_b + y_c^T C         \\
a + s_a - x                           \\
x + s_b - b                           \\
C x + c + s_c                         \\
D(s_a) D(y_a) 1_m                     \\
D(s_b) D(y_b) 1_m                     \\
D(s_c) D(y_c) 1_m
\end{array}
\right)
\] $$
where
$latex x \in \B{R}^n$$,
$latex y_a, s_a \in \B{R}_+^n$$,
$latex y_b, s_b \in \B{R}_+^n$$ and
$latex y_c, s_c \in \B{R}_+^m$$.
The KKT conditions for a solution of this problem is
$latex \[
	F ( x , y_a, s_a, y_b, s_b, y_c, s_c ) = 0
\] $$

$children%example/abs_normal/qp_box.cpp
	%example/abs_normal/qp_box.omh
%$$
$head Example$$
The file $cref qp_box.cpp$$ contains an example and test of
$code qp_box$$.
It returns true if the test passes and false otherwise.

$end
-----------------------------------------------------------------------------
*/
# include "qp_interior.hpp"

// BEGIN C++
namespace CppAD { // BEGIN_CPPAD_NAMESPACE

// BEGIN PROTOTYPE
template <class Vector>
bool qp_box(
	size_t        level   ,
	const Vector& a       ,
	const Vector& b       ,
	const Vector& c       ,
	const Vector& C       ,
	const Vector& g       ,
	const Vector& G       ,
	double        epsilon ,
	size_t        maxitr  ,
	const Vector& xin     ,
	Vector&       xout    )
// END PROTOTYPE
{	double inf = std::numeric_limits<double>::infinity();
	//
	size_t n = a.size();
	size_t m = c.size();
	//
	CPPAD_ASSERT_KNOWN(level <= 2, "qp_interior: level is greater than 2");
	CPPAD_ASSERT_KNOWN(
		size_t(b.size()) == n, "qp_box: size of b is not n"
	);
	CPPAD_ASSERT_KNOWN(
		size_t(C.size()) == m * n, "qp_box: size of C is not m * n"
	);
	CPPAD_ASSERT_KNOWN(
		size_t(g.size()) == n, "qp_box: size of g is not n"
	);
	CPPAD_ASSERT_KNOWN(
		size_t(G.size()) == n * n, "qp_box: size of G is not n * n"
	);
	if( level > 0 )
	{	std::cout << "start qp_box\n";
		CppAD::abs_print_mat("a", n, 1, a);
		CppAD::abs_print_mat("b", n, 1, b);
		CppAD::abs_print_mat("c", m, 1, c);
		CppAD::abs_print_mat("C", m, n, C);
		CppAD::abs_print_mat("g", 1, n, g);
		CppAD::abs_print_mat("G", n, n, G);
		CppAD::abs_print_mat("xin", n, 1, xin);
	}
	//
	// count number of lower and upper limits
	size_t n_limit = 0;
	for(size_t j = 0; j < n; j++)
	{	CPPAD_ASSERT_KNOWN(G[j * n + j] >= 0.0, "qp_box: G_{j,j} < 0.0");
		if( -inf < a[j] )
			++n_limit;
		if( b[j] < inf )
			++n_limit;
	}
	//
	// C_int and c_int define the extended constraints
	Vector C_int((m + n_limit) * n ), c_int(m + n_limit);
	for(size_t i = 0; i < size_t(C_int.size()); i++)
		C_int[i] = 0.0;
	//
	// put C * x + c <= 0 in C_int, c_int
	for(size_t i = 0; i < m; i++)
	{	c_int[i] = c[i];
		for(size_t j = 0; j < n; j++)
			C_int[i * n + j] = C[i * n + j];
	}
	//
	// put I * x - b <= 0 in C_int, c_int
	size_t i_limit = 0;
	for(size_t j = 0; j < n; j++) if( b[j] < inf )
	{	c_int[m + i_limit]            = - b[j];
		C_int[(m + i_limit) * n + j]  = 1.0;
		++i_limit;
	}
	//
	// put a - I * x <= 0 in C_int, c_int
	for(size_t j = 0; j < n; j++) if( -inf < a[j] )
	{	c_int[m + i_limit]           = a[j];
		C_int[(m + i_limit) * n + j] = -1.0;
		++i_limit;
	}
	Vector yout(m + n_limit), sout(m + n_limit);
	size_t level_int = 0;
	if( level == 2 )
		level_int = 1;
	bool ok = qp_interior( level_int,
		c_int, C_int, g, G, epsilon, maxitr, xin, xout, yout, sout
	);
	if( level > 0 )
	{	if( level < 2 )
			CppAD::abs_print_mat("xout", n, 1, xout);
		if( ok )
			std::cout << "end q_box: ok = true\n";
		else
			std::cout << "end q_box: ok = false\n";
	}
	return ok;
}

} // END_CPPAD_NAMESPACE
// END C++

# endif
